Transient wave analysis of a cantilever Timoshenko beam subjected to impact loading by Laplace transform and normal mode methods

This study applies two analytical approaches, Laplace transform and normal mode methods, to investigate the dynamic transient response of a cantilever Timoshenko beam subjected to impact forces. Explicit solutions for the normal mode method and the Laplace transform method are presented. The Durbin method is used to perform the Laplace inverse transformation, and numerical results based on these two approaches are compared. The comparison indicates that the normal mode method is more efficient than the Laplace transform method in the transient response analysis of a cantilever Timoshenko beam, whereas the Laplace transform method is more appropriate than the normal mode method when analyzing the complicated multi-span Timoshenko beam. Furthermore, a three-dimensional finite element cantilever beam model is implemented. The results are compared with the transient responses for displacement, normal stress, shear stress, and the resonant frequencies of a Timoshenko beam and Bernoulli–Euler beam theories. The transient displacement response for a cantilever beam can be appropriately evaluated using the Timoshenko beam theory if the slender ratio is greater than 10 or using the Bernoulli–Euler beam theory if the slender ratio is greater than 100. Moreover, the resonant frequency of a cantilever beam can be accurately determined by the Timoshenko beam theory if the slender ratio is greater than 100 or by the Bernoulli–Euler beam theory if the slender ratio is greater than 400.     

An engineering theory for beam bending

Summary A beam bending theory is proposed that is similar to the Timoshenko and Reissner theories but uses different kinematic variables. Under regular end conditions the theory is shown to predict stresses having a relative mean square error proportional to the depth cubed compared with plane stress elasticity solutions. For a cantilever beam the error from irregular end constraints is found to be smaller in the present theory than in those of Timoshenko and Reissner.

---

Eine technische Theorie der Balkenbiegung

übersicht Es wird eine den Theorien von Timoshenko und Reissner analoge Balkentheorie vorgeschlagen, die aber andersartige kinematische Variable enthÄlt. Unter der Voraussetzung regulÄren Randbedingungen ergeben sich aus dieser Theorie Spannungen, deren relativer mittlerer quadratischer Fehler im Vergleich zu Lösungen ebener ElastizitÄtstheorie proportional zur 3. Potenz der Tragerhöhe ist. Für Kragbalken wird der Fehler von nichtregulÄren Randbedingungen in der dargestellten Theorie kleiner als in den Theorien von Timoshenko und Reissner.

     

Non-linear dynamic response of a rotating radial Timoshenko beam with periodic pulse loading at the free-end

Consideration is given to the dynamic response of a Timoshenko beam under repeated pulse loading. Starting with the basic dynamical equations for a rotating radial cantilever Timoshenko beam clamped at the hub in a centrifugal force field, a system of equations are derived for coupled axial and lateral motions which includes the transverse shear and rotary inertia effects, as well. The hyperbolic wave equation governing the axial motion is coupled with the flexural wave equation governing the lateral motion of the beam through the velocity-dependent skew-symmetric Coriolis force terms. In the analytical formulation, Rayleigh-Ritz method with a set of sinusoidal displacement shape functions is used to determine stiffness, mass and gyroscopic matrices of the system. The tip of the rotating beam is subjected to a periodic pulse load due to local rubbing against the outer case introducing Coulomb friction in the system. Transient response of the beam with the tip deforming due to rub is discussed in terms of the frequency shift and non-linear dynamic response of the rotating beam. Numerical results are presented for this vibro-impact problem of hard rub with varying coefficients of friction and the contact-load time. The effects of beam tip rub forces transmitted through the system are considered to analyze the conditions for dynamic stability of a rotating blade with intermittent rub.     

均布荷载作用下悬臂磁电弹性梁的解析解

对磁电弹性平面问题进行了研究，给出了用拟调和位移函数表达的通解，进而以试凑法按平面应力问题推导出了均布荷载作用下悬臂磁电弹性粱的解析解，所得解有易于理解、便于校对、形式统一简洁的特点。本文还将计算结果与压电材料和弹性材料相应结果进行了分析、比较，为验证各种数值计算方法提供了参考依据。     

The effect of transverse normal strain in contact of an orthotropic beam pressed against a circular surface

The contact problem of a straight orthotropic beam pressed onto a rigid circular surface is considered using beam theories that account for transverse shear and transverse normal deformations. The circular nature of the rigid surface emphasizes the difference between Euler Bernoulli theory behavior, where point loads develop at the edge of contact, and the higher order theories that predict non-singular pressure distributions. While Timoshenko beam theory is the simplest theory that addresses this behavior, the prediction of a maximum value of pressure at the edge of contact contradicts the elasticity theory result that contact pressure must drop to zero. Transverse normal strain is therefore introduced, both to study this fundamental discrepancy and to include an important effect in many contact problems. To investigate this effect, higher order beam theories that account for both constant and linear transverse normal strain through the beam thickness are derived using the principle of virtual work. The resulting orthotropic beam theories depend on the bending stiffness (EI), shear stiffness (GA), axial stiffness (EA₁) and transverse normal stiffness (EA₂), which are independent stiffness parameters that can differ by orders of magnitude. The above mentioned contact problem is then solved analytically for these theories, along with the Timoshenko beam model which assumes zero transverse normal strain. The results for different orthotropic materials show that inclusion of transverse normal deformation has a significant effect on the contact pressure solution. Furthermore, the solution using higher order beam theories encompasses the two extremes of a Hertz-like contact pressure when the half contact length is smaller than the thickness of the beam, and the Timoshenko beam theory case when the half contact length is much larger than the thickness. Concerning the behavior of the pressure at the edge of contact, adherence to the boundary conditions required by the principle of virtual work, shows that while the pressure does tend to zero, it does not become zero unless artificially enforced. In this regard the solution for the case of linear strain is better than that for constant strain. All beam solutions are validated with plane elasticity solutions obtained using the commercial finite element software ABAQUS.     

POLYNOMIAL SOLUTIONS TO PIEZOELECTRIC BEAMS (Ⅱ)——ANALYTICAL SOLUTIONS TO TYPICAL PROBLEMS

IntroductionThis paper is a continuation of Ref.[1],in which a series of orthotropic piezoelectricplane problems was solved and the corresponding exact solutions were obtained with the trial-and-error method,on the basis of the general solution expressed …     

双轴载荷作用下源于椭圆孔的分支裂纹的一种边界元分析

利用一种边界元方法来研究双轴载荷作用下无限大板中源于椭圆孔的分支裂纹．该边界元方法由Crouch与Starfied建立的常位移不连续单元和笔者提出的裂尖位移不连续单元构成．在该边界元方法的实施过程中,左、右裂尖位移不连续单元分别置于裂纹的左、右裂尖处,而常位移不连续单元则分布于除了裂尖位移不连续单元占据的位置之外的整个裂纹面及其它边界,文中算例说明本数值方法对计算平面弹性裂纹的应力强度因子是非常有效的。该文对双轴载荷作用下无限大板中源于椭圆孔的分支裂纹的数值结果进一步证实本数值方法对计算复杂裂纹的应力强度因子的有效性,同时该数值结果可以揭示双轴载荷及裂纹体几何对应力强度因子的影响。     

Analytical test functions for free vibration analysis of rotating non-homogeneous Timoshenko beams

In this paper, the governing equations for free vibration of a non-homogeneous rotating Timoshenko beam, having uniform cross-section, is studied using an inverse problem approach, for both cantilever and pinned-free boundary conditions. The bending displacement and the rotation due to bending are assumed to be simple polynomials which satisfy all four boundary conditions. It is found that for certain polynomial variations of the material mass density, elastic modulus and shear modulus, along the length of the beam, the assumed polynomials serve as simple closed form solutions to the coupled second order governing differential equations with variable coefficients. It is found that there are an infinite number of analytical polynomial functions possible for material mass density, shear modulus and elastic modulus distributions, which share the same frequency and mode shape for a particular mode. The derived results are intended to serve as benchmark solutions for testing approximate or numerical methods used for the vibration analysis of rotating non-homogeneous Timoshenko beams.     

POLYNOMIAL SOLUTIONS TO PIEZOELECTRIC BEAMS(Ⅱ)--ANALYTICAL SOLUTIONS TO TYPICAL PROBLEMS

For the orthotropic piezoelectric plane problem, a series of piezoelectric beams is solved and the corresponding analytical solutions are obtained with the trialand-error method on the basis of the general solution in the case of three distinct eigenvalues, in which all displacements, electrical potential, stresses and electrical displacements are expressed by three displacement functions in terms of harmonic polynomials. These problems are cantilever beam with cross force and point charge at free end, cantilever beam and simply-supported beam subjected to uniform loads on the upper and lower surfaces, and cantilever beam subjected to linear electrical potential.     

Elasticity solutions for plane anisotropic functionally graded beams

This paper considers the plane stress problem of generally anisotropic beams with elastic compliance parameters being arbitrary functions of the thickness coordinate. Firstly, the partial differential equation, which is satisfied by the Airy stress function for the plane problem of anisotropic functionally graded materials and involves the effect of body force, is derived. Secondly, a unified method is developed to obtain the stress function. The analytical expressions of axial force, bending moment, shear force and displacements are then deduced through integration. Thirdly, the stress function is employed to solve problems of anisotropic functionally graded plane beams, with the integral constants completely determined from boundary conditions. A series of elasticity solutions are thus obtained, including the solution for beams under tension and pure bending, the solution for cantilever beams subjected to shear force applied at the free end, the solution for cantilever beams or simply supported beams subjected to uniform load, the solution for fixed–fixed beams subjected to uniform load, and the one for beams subjected to body force, etc. These solutions can be easily degenerated into the elasticity solutions for homogeneous beams. Some of them are absolutely new to literature, and some coincide with the available solutions. It is also found that there are certain errors in several available solutions. A numerical example is finally presented to show the effect of material inhomogeneity on the elastic field in a functionally graded anisotropic cantilever beam.     

Explicit frequency equations of free vibration of a nonlocal Timoshenko beam with surface effects

Considerations of nonlocal elasticity and surface effects in micro-and nanoscale beams are both important for the accurate prediction of natural frequency. In this study, the governing equation of a nonlocal Timoshenko beam with surface effects is established by taking into account three types of boundary conditions: hinged–hinged, clamped–clamped and clamped–hinged ends. For a hinged–hinged beam, an exact and explicit natural frequency equation is obtained. However, for clamped–clamped and clamped–hinged beams, the solutions of corresponding frequency equations must be determined numerically due to their transcendental nature. Hence, the Fredholm integral equation approach coupled with a curve fitting method is employed to derive the approximate fundamental frequency equations, which can predict the frequency values with high accuracy. In short,explicit frequency equations of the Timoshenko beam for three types of boundary conditions are proposed to exhibit directly the dependence of the natural frequency on the nonlocal elasticity, surface elasticity, residual surface stress, shear deformation and rotatory inertia, avoiding the complicated numerical computation.     

Analysis of micropolar elastic beams

In this paper, a linear theory for the analysis of beams based on the micropolar continuum mechanics is developed. Power series expansions for the axial displacement and micro-rotation fields are assumed. The governing equations are derived by integrating the momentum and moment of momentum equations in the micropolar continuum theory. Body couples and couple stresses can be supported in this theory. After some simplifications, this theory can be reduced to the well-known Timoshenko and Euler–Bernoulli beam theories. The nature of flexural and longitudinal waves in the infinite length micropolar beam has been investigated. This theory predicts the existence of micro-rotational waves which are not present in any of the known beam theories based on the classical continuum mechanics. Also, the deformation of a cantilever beam with transverse concentrated tip loading has been studied. The pattern of deflection of the beam is similar to the classical beam theories, but couple stress and micro-rotation show an oscillatory behavior along the beam for various loadings.     

Finite versus small strain discrete dislocation analysis of cantilever bending of single crystals

Plastic size effects in single crystals are investi-gated by using finite strain and small strain discrete dislo-cation plasticity to analyse the response of cantilever beam specimens. Crystals with both one and two active slip sys-tems are analysed, as well as specimens with different beam aspect ratios. Over the range of specimen sizes analysed here, the bending stress versus applied tip displacement response has a strong hardening plastic component. This hardening rate increases with decreasing specimen size. The hardening rates are slightly lower when the finite strain discrete disloca-tion plasticity (DDP) formulation is employed as curving of the slip planes is accounted for in the finite strain formulation. This relaxes the back-stresses in the dislocation pile-ups and thereby reduces the hardening rate. Our calculations show that in line with the pure bending case, the bending stress in cantilever bending displays a plastic size dependence. How-ever, unlike pure bending, the bending flow strength of the larger aspect ratio cantilever beams is appreciably smaller. This is attributed to the fact that for the same applied bend-ing stress, longer beams have lower shear forces acting upon them and this results in a lower density of statistically stored dislocations.     

A micro scale Timoshenko beam model based on strain gradient elasticity theory

A micro scale Timoshenko beam model is developed based on strain gradient elasticity theory. Governing equations, initial conditions and boundary conditions are derived simultaneously by using Hamilton's principle. The new model incorporated with Poisson effect contains three material length scale parameters and can consequently capture the size effect. This model can degenerate into the modified couple stress Timoshenko beam model or even the classical Timoshenko beam model if two or all material length scale parameters are taken to be zero respectively. In addition, the newly developed model recovers the micro scale Bernoulli–Euler beam model when shear deformation is ignored. To illustrate the new model, the static bending and free vibration problems of a simply supported micro scale Timoshenko beam are solved respectively. Numerical results reveal that the differences in the deflection, rotation and natural frequency predicted by the present model and the other two reduced Timoshenko models are large as the beam thickness is comparable to the material length scale parameter. These differences, however, are decreasing or even diminishing with the increase of the beam thickness. In addition, Poisson effect on the beam deflection, rotation and natural frequency possesses an interesting “extreme point” phenomenon, which is quite different from that predicted by the classical Timoshenko beam model.     

A homogenization-based theory for anisotropic beams with accurate through-section stress and strain prediction

This paper presents a homogenization-based theory for three-dimensional anisotropic beams. The proposed beam theory uses a hierarchy of solutions to carefully-chosen beam problems that are referred to as the fundamental states. The stress and strain distribution in the beam is expressed as a linear combination of the fundamental state solutions and stress and strain residuals that capture the parts of the solution not accounted for by the fundamental states. This decomposition plays an important role in the homogenization process and provides a consistent method to reconstruct the stress and strain distribution in the beam in a post-processing calculation. A finite-element method is presented to calculate the fundamental state solutions. Results are presented demonstrating that the stress and strain reconstruction achieves accuracy comparable with full three-dimensional finite element computations, away from the ends of the beam. The computational cost of the proposed approach is three orders of magnitude less than the computational cost of full three-dimensional calculations for the cases presented here. For isotropic beams with symmetric cross-sections, the proposed theory takes the form of classical Timoshenko beam theory with Cowper’s shear correction factor and additional load-dependent corrections. The proposed approach provides an extension of Timoshenko’s beam theory that handles sections with anisotropic construction.     

自由端受集中力作用下压电悬臂梁弯曲问题解析解

本文对由横观各向同性压电介质构成的悬臂梁，在自由端受集中力作用下的弯曲问题进行了研究。首先根据问题的特点，得到简化的线弹性压电悬臂梁的基本方程。然后根据正交各向异性材料悬臂梁应力分布特点，采用逆解法，建立了该问题的应力函数与电势分布函数，进而得到精确多项式解析解。该解析解形式简单，便于应用。文中对自由端受集中力的常规材料和压电材料悬臂梁的挠度也进行了比较。     

On anti-plane shear behavior of a Griffith permeable crack in piezoelectric materials by use of the non-local theory

In this paper, the non-local theory of elasticity is applied to obtain the behavior of a Griffith crack in the piezoelectric materials under anti-plane shear loading for permeable crack surface conditions. By means of the Fourier transform the problem can be solved with the help of a pair of dual integral equations with the unknown variable being the jump of the displacement across the crack surfaces. These equations are solved by the Schmidt method. Numerical examples are provided. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularity is present at the crack tip. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum stress hypothesis. The finite hoop stress at the crack tip depends on the crack length and the lattice parameter of the materials, respectively. The project supported by the National Natural Science Foundation of China (50232030 and 10172030)     

Modeling and bifurcation analysis of the centre rigid-body mounted on an external Timoshenko beam

Ａｓｗｅｋｎｏｗ,ｉｔｉｓｔｈｅｃｈａｒａｃｔｅｒｉｓｔｉｃｏｆｔｈｅｆｌｅｘｉｂｌｅｍｕｌｔｉｂｏｄｙｓｙｓｔｅｍｔｈａｔｔｈｅｒｉｇｉｄｍｏｔｉｏｎｉｓｃｏｕｐｌｅｄｗｉｔｈｔｈｅｆｌｅｘｉｂｌｅｄｅｆｏｒｍａｔｉｏｎ［１～３］．Ｔｈｅｒｅｆｏｒｅ,ｆｏｒｔｈｅｆｌｅｘｉｂｌｅｍｕｌｔｉｂｏｄｙｓｙｓｔｒｍ,ｉｔｉｓｎｅｃｅｓｓａｒｙｔｏｉｎｖｅｓｔｉｇａｔｅｔｈｅｒｅｇｕｌａｒｏｆｔｈｅｄｅｓｔａｂｉｌｉｚａｔｉｏｎａｎｄｂｉｆｕｒｃａｔｉｏｎｏｆｉｔｓｃｏｎｆｉｇｕｒａｔｉｏｎｂｅｆｏｒ…     

Analytical solution to a mixed boundary value elastic problem of a roller-guided panel of laminated composite

This study presents an analytical solution to elastic field in a roller-guided panel of symmetric cross-ply laminated composite material. The mixed boundary value two-dimensional plane stress elasticity problem is formulated in terms of a single displacement potential function. This reduces the problem to the solution of a single fourth order partial differential equation of equilibrium as the other equilibrium equation is satisfied automatically. The solution is obtained in terms of an infinite Fourier series. To present some numerical results, a panel of glass/epoxy laminated composite is considered and different components of stress and displacement at different sections of the panel are presented graphically. To justify the present analytical solution, it is compared with the finite element solution obtained by using the commercial software ANSYS. It is found that the two solutions agree well with each other. This ensures that the formulation developed in this study based on the displacement potential approach can be used to obtain analytical solution of an elastic field in structural elements of laminated composite under any mode of boundary conditions prescribed in terms of either stress, displacement or any combination of these.